The multiplicative complexity of a Boolean function is the minimum number of two-input AND gates that are necessary and sufficient to implement the function over the basis (AND, XOR, NOT). Finding the multiplicative complexity of a given function is computationally intractable, even for functions with small number of inputs. Turan et al. [1] showed that n-variable Boolean functions can be implemented with at most n-1 AND gates for n ≤ 5. A counting argument can be used to show that, for n ≥ 7, there exist n-variable Boolean functions with multiplicative complexity of at least n. In this work, we propose a method to find the multiplicative complexity of Boolean functions by analyzing circuits with a particular number of AND gates and utilizing the affine equivalence of functions. We use this method to study the multiplicative complexity of 6-variable Boolean functions, and calculate the multiplicative complexities of all 150357 affine equivalence classes. We show that any 6-variable Boolean function can be implemented using at most 6 AND gates. Additionally, we exhibit specific 6-variable Boolean functions which have multiplicative complexity 6.
Keywords: 06E30; 94A60; Affine equivalence; Boolean functions; Circuit complexity; Cryptography; Multiplicative complexity.